Notice that this transformation is different to the plain pull-back considered in [1]}, [2]}, which is induced by the given diffeomorphism of the domains.
The reason for using this map is that we want to ensure that the transformation takes divergence free functions to divergence free functions.
This transformation is known as the (contravariant) Piola transform in the particular case that \(\det (D\Phi _t)\)  is spacially constant, for more details see [3]}, for example.
With this definition in mind and the appropriate regularity of \(\Phi _t\) , courtesy of Lemma REF , we give the result that \(\phi _t\)  and \(\phi _{-t}\)  preserve the divergence of the functions they are applied to.
